How to keep the query filters sent via $ _POST?

Just as @Bacco commented, you can have multiple filters on different tabs, that is, using the same session, you can use the GET filter, which is quite plausible.

<?php
   $filtro = '';
   if (isset($_REQUEST['imovel'])){ // REQUEST busca os dados tanto de POST como de GET
     $imovel = $_REQUEST['imovel']; // Trate o SQL Injection
     $where = " imovel = {$imovel} AND "
     $filtro .= '&imovel='.urlencode($imovel);
   }

     $page = ((isset($_GET['pagina']) && intval($_GET['pagina']) > 1)? intval($_GET['pagina']) : 1) - 1;
     $limite = 12;
     $atual = $page * $limite;
     $limit = " LIMIT {$atual}, {$limite}";



     // Registros limitados
     $query    = $pdo->query("SELECT * FROM imovel ".trim($where, ' AND ').$limit); 
     $total    = $pdo->query("SELECT * FROM imovel ".trim($where, ' AND '));
     $total2   = $total->rowCount();
     $qtdpage  = ceil($total2/$limite); // Quantidade de páginas

     for ($i = 1; $i < $qtdpage; $i++){
          echo '<li><a href="busca?pagina='.$i.$filtro.'">'.$i.'</a></li>';
     }

Or store query data in sessions:

<?php
   session_start();
   if (isset($_POST['imovel'])){
     $_SESSION['filtro'] = Array();
     // Não se esquça de fazer tratamento para SQL Injection
     $_SESSION['filtro']['imovel'] = $_POST['imovel']; 
   }

     $page = ((isset($_GET['pagina']) && intval($_GET['pagina']) > 1)? intval($_GET['pagina']) : 1) - 1;
     $limite = 12;
     $atual = $page * $limite;
     $limit = " LIMIT {$atual}, {$limite}";

     if (isset($_SESSION['filtro']['imovel'])){
        $where = " imovel = {$_SESSION['filtro']['imovel']} AND "
     }

     // Registros limitados
     $query    = $pdo->query("SELECT * FROM imovel ".trim($where, ' AND ').$limit); 
     $total    = $pdo->query("SELECT * FROM imovel ".trim($where, ' AND '));
     $total2   = $total->rowCount();
     $qtdpage  = ceil($total2/$limite); // Quantidade de páginas

     for ($i = 1; $i < $qtdpage; $i++){
          echo '<li><a href="busca?pagina='.$i.'">'.$i.'</a></li>';
     }

If I understand correctly you lose the values of the query and to keep the values as you go through POST simply load them in input of the query, the example is:

echo "<input name='consulta'";
if($consulta = filter_input(INPUT_POST , 'consulta'))
    echo "value='$consulta'"; 
echo ">";